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arXiv · 2401.05003

Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori

Joan Gimeno·Àngel Jorba·Begoña Nicolás·Estrella Olmedo
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Abstract

In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in stroboscopic Poincar\'e maps. The method has two steps. In the first step we compute, by means of a quadratically convergent scheme, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. If the torus has stable and/or unstable directions, in the second step we compute the Taylor-Fourier expansions of the corresponding invariant manifolds up to a given order. The paper also discusses the case in which the torus is highly unstable so that a multiple shooting strategy is needed to compute the torus. If the order of the Taylor expansion of the manifolds is fixed and N is the number of Fourier modes, the whole computational effort (torus and manifolds) increases as O(N log N) q and the memory required behaves as O(N). This makes the algorithm very suitable to compute high-dimensional tori for which a huge number of Fourier modes are needed. Besides, the algorithm has a very high degree of parallelism. The paper includes examples where we compute invariant tori (of dimensions up to 5) of quasi-periodically forced ODEs. The computations are run in a parallel computer and its efficiency with respect to the number of processors is also discussed.

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Joan Gimeno, Àngel Jorba, Begoña Nicolás, Estrella Olmedo. 2024-01-10. Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori. https://doi.org/10.1137/21m1458363

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